MatNoble/PDE_coding

FDM, FEM

MATLAB

Some Results

Landau–Lifshitz equation

Marker and Cell Methond for Stokes Equations

Using FDM or FEM solve poisson and heat equations in 1D and 2D

Basic equations

poisson equation:

$-\Delta u = f$

heat equation:

$u_t - \Delta u = f$

Finite Difference Methods

Replacing the dervatives by finite differences(to do finite difference scheme)

$u^k_t = \frac{u^{k+1}-u^k}{\Delta t}$

$\begin{aligned} u_{x x} & \approx \frac{u\left(x_{i-1}, y_{j}\right)-2 u\left(x_{i}, y_{j}\right)+u\left(x_{i+1}, y_{j}\right)}{h_{x}^{2}} \\ u_{y y} & \approx \frac{u\left(x_{i}, y_{j-1}\right)-2 u\left(x_{i}, y_{j}\right)+u\left(x_{i}, y_{j+1}\right)}{h_{y}^{2}} \end{aligned}$
Taylor series(to do error estimate)

$f(x)=\frac{f\left(x_{0}\right)}{0 !}+\frac{f^{\prime}\left(x_{0}\right)}{1 !}\left(x-x_{0}\right)+\ldots+\frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}+R_{n}(x)$

$R_{n}(x)=\mathrm{o}\left[\left(x-x_{0}\right)^{n}\right]$

Example 1

for the 1D heat problem:

$\left\{\begin{array}{c}{u_{t}- u_{x x}=f(x, t)\left(x_{1}<x<x_{2}\right)(t>0)} \\ {u(x, 0)=g_{t}(x)\left(x_{1}<x<x_{2}\right)} \\ {u\left(x_{1}, t\right)=g_{1}(t), u\left(x_{2}, t\right)=g_{2}(t)(t>0)}\end{array}\right.$

with specific parameters:

$0.1<x<\pi,\ 0<t<0.2$

$u(x, 0)=10 \frac{\cos \left(\left(\frac{x-x_{1}}{2}\right)^{2}\right) \sin \left(\left(x-x_{2}\right)^{3}\right)}{x^{\frac{4}{5}}}\\$

$u\left(x_{1}, t\right)=u\left(x_{2}, t\right)=0$

have the flowing result:

Finite Element Methods

Integration by part
Variational
Weak formula
Garlekin formula
Finite dimensional space approximates infinite dimensional space

Approximate functions

Gibbs phenomenon